(1)

The table below shows
all the whole numbers
as the base numbers &
the respective triangular numbers
generated from these.

Triangular Numbers Table
Base
number
Triangular
number

•••••



••
•••
••••
•••••

0

(0 x 1)/2 = 0

1

(1 x2 )/2 = 1

4

(4 x 5)/2 = 10

9

(9 x 10)/2 = 45

16

(16 x 17)/2 = 136

(2)

The tabulation below shows
all the sub-totals of
the numbers per (1).

0 = 0
0 + 1 = 1
0 + 1 + 10 = 11
0 + 1 + 10 + 45 = 56 
0 + 1 + 19 + 45 + 136 = 192




(3)

The tabulation below shows
the numbers per (2)
multiplied by 60
& the resulting products
increased by
respective uneven numbers
starting from 1. 

60(0) + 1 = 1
60(1) + 3 = 63
60(11) + 5 = 665
60(56) + 7 = 3367 
60(192) + 9 = 11529




(4)

The tabulation below shows
all the sub-totals of
the numbers per (3),
resulting in all whole numbers
to the 6th. power.

1 = 1 = 16
1 + 63 = 64 = 26
1 + 63 + 665 = 729 = 36
1 + 63 + 665 + 3367 = 4096 = 46
1 + 63 + 665 + 3367 + 11529 = 15625 = 56




From the tabulation above
one can deduce that in general
n to the 6th. power is
the summation of triangular numbers 
as shown below:

n4 =

n-1
Σ
k=0

n-1
[Σ
k=0

[30k2(k6 + 1) + (2k + 1)]

]